Question: Divide the following complex numbers. $ \dfrac{7+i}{-4+3i}$
Solution: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-4-3i}$ $ \dfrac{7+i}{-4+3i} = \dfrac{7+i}{-4+3i} \cdot \dfrac{{-4-3i}}{{-4-3i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(7+i) \cdot (-4-3i)} {(-4+3i) \cdot (-4-3i)} = \dfrac{(7+i) \cdot (-4-3i)} {(-4)^2 - (3i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(7+i) \cdot (-4-3i)} {(-4)^2 - (3i)^2} = $ $ \dfrac{(7+i) \cdot (-4-3i)} {16 + 9} = $ $ \dfrac{(7+i) \cdot (-4-3i)} {25} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({7+i}) \cdot ({-4-3i})} {25} = $ $ \dfrac{{7} \cdot {(-4)} + {1} \cdot {(-4) i} + {7} \cdot {-3 i} + {1} \cdot {-3 i^2}} {25} $ Evaluate each product of two numbers. $ \dfrac{-28 - 4i - 21i - 3 i^2} {25} $ Finally, simplify the fraction. $ \dfrac{-28 - 4i - 21i + 3} {25} = \dfrac{-25 - 25i} {25} = -1-i $